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Enumeration of
symmetric arc diagrams
Juan B. Gil and Luis E. Lopez |
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Vol. 16 (2023), No. 1, 107–126
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Abstract |
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We give recurrence relations for the enumeration of symmetric elements within four classes of arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These arc diagrams are motivated by the study of RNA secondary structures. For example, classic RNA secondary structures correspond to 3412-avoiding involutions with no adjacent transpositions, and structures with base triples may be represented as partitions with crossings. Our results rely on combinatorial arguments. In particular, we use Motzkin paths to describe noncrossing arc diagrams that have no arc connecting two adjacent nodes, and we give an explicit bijection to ternary words whose length coincides with the sum of their digits. We also discuss the asymptotic behavior of some of the sequences considered here in order to quantify the extremely low probability of finding symmetric structures with a large number of nodes. |
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Keywords
symmetric arc diagrams, secondary structures,
pattern-avoiding involutions, noncrossing partitions
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Mathematical Subject Classification
Primary: 05A19
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Milestones
Received: 20 September 2021
Revised: 19 March 2022
Accepted: 23 March 2022
Published: 14 April 2023
Communicated by Arthur T. Benjamin
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Authors |
| Juan B. Gil | |
| Penn State Altoona Altoona, PA United States |
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| Luis E. Lopez | |
| Penn State Altoona Altoona, PA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). |
